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Unifying model of shoot gravitropism reveals proprioception as a central feature ofposture control in plants.

Bastien, Renaud; Bohr, Tomas; Moulia, Bruno; Douady, Stéphane

Published in:Proceedings of the National Academy of Sciences of the United States of America

Publication date:2012

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Citation (APA):Bastien, R., Bohr, T., Moulia, B., & Douady, S. (2012). Unifying model of shoot gravitropism revealsproprioception as a central feature of posture control in plants. Proceedings of the National Academy ofSciences of the United States of America, 110(2), 755-760.

Unifying model of shoot gravitropism revealsproprioception as a central feature ofposture control in plantsRenaud Bastiena,b,c, Tomas Bohrd, Bruno Mouliaa,b,1,2, and Stéphane Douadyc,2

aINRA (Institut National de la Recherche Agronomique), UMR0547 (Unité Mixte de Recherche PIAF Physique et Physiologie Intégratives de l’Arbre Fruitier etForestier), F-63100 Clermont-Ferrand, France; bClermont Université, Université Blaise Pascal, UMR0547 (Unité Mixte de Recherche PIAF Physique et PhysiologieIntégratives de l’Arbre Fruitier et Forestier), BP 10448, F-63000 Clermont-Ferrand, France; cMatière et Systèmes Complexes, Université Paris-Diderot, 75025Paris Cedex 13, France; and dDepartment of Physics and Center for Fluid Dynamics, Technical University of Denmark, DK-2800 Lyngby, Denmark

Edited by Przemyslaw Prusinkiewicz, University of Calgary, Calgary, AB, Canada, and accepted by the Editorial Board November 2, 2012 (received for reviewAugust 17, 2012)

Gravitropism, the slow reorientation of plant growth in responseto gravity, is a key determinant of the form and posture of landplants. Shoot gravitropism is triggered when statocysts sense thelocal angle of the growing organ relative to the gravitationalfield. Lateral transport of the hormone auxin to the lower side isthen enhanced, resulting in differential gene expression and cellelongation causing the organ to bend. However, little is knownabout the dynamics, regulation, and diversity of the entirebending and straightening process. Here, we modeled the bendingand straightening of a rod-like organ and compared it with thegravitropism kinematics of different organs from 11 angiosperms.We show that gravitropic straightening shares common traitsacross species, organs, and orders of magnitude. The minimaldynamic model accounting for these traits is not the widely citedgravisensing law but one that also takes into account the sensingof local curvature, what we describe here as a graviproprioceptivelaw. In our model, the entire dynamics of the bending/straight-ening response is described by a single dimensionless “bendingnumber” B that reflects the ratio between graviceptive and pro-prioceptive sensitivities. The parameter B defines both the finalshape of the organ at equilibrium and the timing of curving andstraightening. B can be estimated from simple experiments, andthe model can then explain most of the diversity observed inexperiments. Proprioceptive sensing is thus as important as grav-isensing in gravitropic control, and the B ratio can be measured asphenotype in genetic studies.

perception | signaling | movement | morphogenesis

Plant gravitropism is the growth movement of organs in re-sponse to gravity that ensures that most shoots grow up and

most roots grow down (1–6). As for all tropisms, a directionalstimulus is sensed (gravity in this case), and the curvature of theorgan changes over time until a set-angle and a steady-state shapeare reached (2, 7, 8). The change in shape is achieved by differ-ential elongation for organs undergoing primary growth (e.g.,coleoptiles) or by differential differentiation and shrinkage of re-action wood for organs undergoing secondary growth (e.g., treetrunks) (9). Tropisms are complex responses, as unlike other plantmovements (e.g., fast movements) (5, 10) the motor activity gen-erated is under continuous biological control (e.g., refs. 3, 11, 12).The biomechanics of plant elongation growth has been ana-

lyzed in some detail (5, 13, 14), but less is known about the bi-ological control of tropic movements and differential growth (3,6). Many molecular and genetic processes that occur insidesensing and motor cells have been described (2, 15). For exam-ple, statocysts are cells that sense gravity through the complexmotion of small intercellular bodies called statoliths (16). How-ever, a huge number of sensing and motor cells act together toproduce the growth movements of a multicellular organ. Howare the movements of an organ controlled and coordinated bi-ologically? This is a key question, as establishing the correctposture of aerial organs with respect to the rest of the plant has

important physiological and ecological consequences (e.g., ac-cess to light or long-term mechanical stability) (4).The gravitropic responses of some plants and even fungi have

similar features (8). In essence, this has been described as a bi-phasic pattern of general curving followed by basipetal straight-ening (GC/BS) (4, 17). First, the organ curves up gravitropically,then a phase of decurving starts at the tip and propagates down-ward, so that the curvature finally becomes concentrated at thebase of the growth zone and steady (7–9, 18–20). This decurving,which has also been described as autotropic (i.e., the tendency ofplants to recover straightness in the absence of any externalstimulus) (7, 21), may start before the tip reaches the vertical (4).It is striking that organs differing in size by up to four orders ofmagnitude (e.g., from an hypocotyl to the trunk of an adult tree)display similar traits, despite great differences in the timing of thetropic movement and the motor processes involved (3). However,there are also differences in the gravitropic responses. Dependingon the species and the growth conditions, plants may or may notoscillate transiently about the stimulus axis or reach a properalignment with the direction of the stimulus (e.g., ref. 8).Currently, the phenotypic variability of the GC/BS biphasic

pattern over a broad sample of species is, however, hard to es-timate quantitatively, as most studies of gravitropism have onlyfocused on measuring the tip angle (3). As we shall demonstrate,it is necessary to specify the local curvature C (or equivalently,the inclination angle A) over the entire growth zone (Fig. 1) andhow it changes over time. If this is done, it is possible to build upa minimal dynamic model for tropic movements in space. Thiscan be combined with dimensional analysis (as is used in fluidmechanics, for example) to characterize the size and time de-pendencies and set up dimensionless control parameters. Thisthen makes it possible to compare experiments with predictionsfrom the model quantitatively over a broad taxonomical sampleof species with very different sizes and growth velocities and toreveal universal behaviors and controlling mechanisms.The gravitropic responses of 12 genotypes from 11 plant

species were studied, representing a broad taxonomical range ofland angiosperms (SI Appendix, Fig. S4), major growth habits(herbs, shrubs, and trees), as well as different uses (agriculture,horticulture, and forestry but also major laboratory model plantsfor genetics and physiology). Different types of organs werestudied: coleoptile, hypocotyl, epicotyl, herbaceous and woody

Author contributions: B.M. and S.D. designed research and hypotheses; R.B. developed themodels, the experiments, and the numerical simulations; R.B. and T.B. solved the equations;R.B., B.M., and S.D. analyzed data; and R.B., T.B., B.M., and S.D. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. P.P. is a guest editor invited by the EditorialBoard.1To whom correspondence should be addressed. E-mail: bruno.moulia@clermont.inra.fr.2B.M. and S.D. contributed equally to this work.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1214301109/-/DCSupplemental.

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vegetative stems, and inflorescence stems, representing the twotypes of tropic motors (differential elongation growth, reactionwoods) and varying by two orders of magnitude in organ size andin the timing of the tropic movements. Organs were tilted hori-zontally and the gravitropic growth was recorded through time-lapse photography.All of the plant organs studied first curved upwards before

eventually reaching a near vertical steady-state form where theapical part was straight, as shown for two examples in Fig. 1 andin Movies S1 and S2. The images were used to generate colormaps of the curvature of the organ in space (along the organ)and time, as shown for three examples in Fig. 2. Shortly afterplants were placed horizontally, the dominant movement ob-served was a rapid up-curving (negatively gravitropic) along theentire organ. However, the apex soon started to straighten andthe straightening gradually moved downward along the organ.Finally, the curvature tended to concentrate at the base of thegrowth zone, becoming fixed there. Such a typical GC/BS be-havior was observed in all 12 cases studied, despite differences ofaround two orders of magnitude in organ sizes and convergence

time, the time Tc taken for the organ to return to a steady state,ranging from several hours to several months.Despite the common properties of the response, time lapse

photography showed that plant organs acted differently whenapproaching the vertical. The apices of some plant organs neverovershot the vertical (Fig. 1A), whereas others did so several times,exhibiting transient oscillations with the formation of C- or even S-shapes (Fig. 1B). Thus, a minimal dynamic model of gravitropismhas to explain both the common biphasic GC/BS pattern and thediversity in transient oscillation and convergence time.According to the literature, the current qualitative model

of gravitropism in aerial shoots is based on the followinghypotheses:

H1: Gravisensing is exclusively local; each element along thelength of the organ is able to respond to its current state(22), since statocysts are found all along the growth zone(16). Gravisensing by the apex does not have a special in-fluence (e.g., the final shapes of organs after decapitationare similar to intact controls) (1, 23).

H2: The local inclination angle A (Fig. 1) is sensed. This sensingfollows a sine law (3, 6) (see below).

H3: In our reference frame, the so-called gravitropic set angle(GSA) (24) is equal to 0 (Fig. 1) so the motion tends tobring the organ upward toward the vertical (this corre-sponds to the botanical term “negative ortho-gravitropism,”a most common feature in shoots).

H4: The action of the tropic motor is fully driven by the percep-tion–regulation process and results in a change in the localcurvature through differential growth and/or tissue differ-entiation. This response can only be expressed where dif-ferential growth and differentiation occurs, namely in the“growth zone” of length Lgz (3).

To form a mathematical model, we shall describe the shape ofthe organ in terms of its median—that is, its central axis (Fig. 1).We parameterize the median by the arc length s going from thebase s= 0 to the apex s=L, and the angle Aðs; tÞ describes thelocal orientation of the median with respect to the vertical attime t. The corresponding local curvature Cðs; tÞ is the spatialrate of change of A along s and from differential geometry weknow that:

Cðs; tÞ= ∂Aðs; tÞ=∂s  or  Aðs; tÞ= A0 +Zs

0

Cðl; tÞdl: [1]

The so-called “sine law” was first defined by Sachs in the 19thcentury and has been widely used since (see ref. 3 for a review).It can be expressed as a relationship between the change in thelocal curvature and the local angle as in:

∂Cðs; tÞ=∂t = −β  sin Aðs; tÞ; [2]

where β is the apparent gravisensitivity. Note that Eq. 2 is un-changed when A changes to −A and C changes to −C, as wouldbe expected. This model is only valid in the growth zone,s> ðL−LgzÞ, where L is the total organ length and Lgz is thelength of the effective zone where active curving can be achieved.Outside this region, the curvature does not change with time.In this model, changes in the overall length of the organ are

not taken into account. This is quite reasonable in the case ofwoody organs, as they undergo curving through relatively smallmaturation strains in reaction woods, but it is less applicable toorgans curving through differential elongation (3, 14). In ex-panding organs, each segment of the organ in the growth zone“flows” along the organ being pushed by the expansion growthof distal elements (3, 14) so Eq. 2 would remain valid only in

A

B

s=0

s=L

x

y

A(s)

s=L-L

Lgz

gz

s

s+dsdA

C

Fig. 1. Successive shapes formed by plant organs undergoing gravitropismand a geometrical description of these shapes. (A) Time-lapse photographsof the gravitropic response of a wheat coleoptile placed horizontally (MovieS1). (B) Time-lapse photographs of the gravitropic response of an Arabi-dopsis inflorescence placed horizontally (Movie S2). White bars, 1 cm. (C)Geometric description of organ shape. The median line of an organ of totallength L is in a plane defined by coordinates x, y. The arc length s is definedalong the median line, with s = 0 referring to the base and s = L referring tothe apex. In an elongating organ, only the part inside the growth zone oflength Lgz from the apex is able to curve (with Lgz = L at early stages andLgz < L later on), whereas the whole length is able to curve in organs un-dergoing secondary growth (i.e., Lgz = L). AðsÞ is the local orientation of theorgan with respect to the vertical and CðsÞ the local curvature. The twocurves shown have the same apical angle AðLÞ but different shapes, so tospecify the shape we need the form of AðsÞ or CðsÞ along the entire median.Due to the symmetry of the system around the vertical axis, the angle A isa zenith angle—that is, it is zero when the organ is vertical and upright.Thus, an orthotropic organ has a gravitropic set point angle of 0. For sim-plicity, clockwise angles are considered as positive.

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a “comoving” context. To fully specify the changes in curvature,we would thus have to introduce local growth velocities into themodel, replacing the derivative in Eq. 2 with the comoving de-rivative DCðs; tÞ=Dt= ∂Cðs; tÞ=∂t+ vðs; tÞ∂Cðs; tÞ=∂s, where vðs; tÞis the local growth velocity. However, in tropic movement, thegrowth velocities are generally small compared with tropicbending velocities (and the length of the organ that has left thegrowth zone during the straightening movement is also small)(14), so DCðs; tÞ=Dt≈∂Cðs; tÞ=∂t. The limits of this approxima-tion will be discussed.To obtain a more tractable model, which we shall solve ana-

lytically, we can use the approximation sin  A ≈ A+OðA3Þ andapproximate Eq. 2 by:

∂Cðs; tÞ=∂t = −βAðs; tÞ; [3]

where we note that the A→ −A symmetry is retained. Becausein our experiments jAj did not exceed π=2 and because we areprimarily interested in values near zero, this is a reasonableapproximation (3). It should be noted that Aðs; tÞ and Cðs; tÞ arenot independent, as any further variation in curvature modifiesthe apical orientation through the “lever-arm effect” expressedin Eq. 1. In other words, the effect of changes in curvature ondownstream orientation angles is amplified by the distance alongthe organ (3).The solution of Eq. 3, which we shall call the “A model,” is:

A = A0J0�2

ffiffiffiffiffiffiβts

p �;      C = A0

ffiffiffiffiβts

rJ1�2

ffiffiffiffiffiffiβts

p �; [4]

where Jn are Bessel functions of the first kind of order n. It hasinteresting properties. Firstly, the angle A does not depend onspace s and time t individually, but only on the combination offfiffiffiffits

pand

ffiffiffiffiffiffit=s

pand is thus an oscillatory function of

ffiffiffiffits

p. However,

the dynamics of the A model demonstrates that such a systemcannot reach a vertical steady state when tilted and clamped atits base (Fig. 3A and Movie S3). Indeed, the only steady state inEq. 3 is Aðs; tÞ= 0, but this is forbidden by the basal clamping ofthe organ fixing Aðs= 0; tÞ= π=2 for all t. Oscillations thereforego on indefinitely, whereas their wavelengths decrease with time.Numerical simulations of Eqs. 3 or 2 displayed the same behav-ior (SI Appendix Fig. S2). This does not agree with any of theexperimental results. The A model based on the sine law istherefore not a suitable dynamic model of the gravitropicstraightening movement and has to be rejected. To account forthe steady state attained after tilting, another hypothesis needs tobe introduced:

H5: Each constituent element of the organ perceives its localdeformation, the curvature, and responds in order to restorelocal straightness (7, 19). In animal physiology, this type ofsensing is generally called “proprioception,” a self-sensingof posture or orientiation of body parts relative to the rest ofthe organism (25). This is not an unreasonable assumptionas it is known experimentally that (i) plants can sense im-posed bending (26, 27) and (ii) the curvature of the organand subsequent mechanical loads have a direct effect on

s(mm)

t(h

)

0 10 20 30

0

5

10

15

20

0

0.05

0.1

0.15

s(mm)

t(h

)

0 20 40 60 80

0

5

10

15

20

0

0.02

0.04

0.06

0.08

0.1

s(cm)

t(w

eek)

0 100 200

0

2

4

6

8

10

0

1

2

3x 10

-3A B C

C(m

m ) -1

C(m

m ) -1

C(m

m ) -1

Fig. 2. Kinematics of the entire tropic movement of tilted plant organs shown as color maps plotting the curvature Cðs; tÞwith respect to time t and curvilinearabscissa s (the arc length along the median measured from the base to apex of the organ; Fig. 1). (A) Wheat coleoptile (Triticum aestivum cv. Recital). Theyellow bar is 1 cm long. (B) Arabidopsis inflorescence (A. thaliana ecotype Col0). The yellow bar is 1 cm long. (C) Poplar trunk (Hybrid Populus deltoides x nigracv I4551), reprocessed data from ref. 9.

-0.2 0 0.2 0.4 0.6 0.8

0

0.2

0.4

0.6

0.8

1

x/L

y/L

s/L

tL C

L

0 0.2 0.4 0.6 0.8

0

10

20

30

40

50

-0.2 0 0.2 0.4 0.6 0.8

0

0.2

0.4

0.6

0.8

1

x/L

y/L

s/L

tL

CL

0 0.2 0.4 0.6 0.8

0

10

20

30

40

50

-10

-5

0

5

10

0

2.5

5

7.5

10

A

B

Fig. 3. Solutions of the dimensionless A and AC models. (Left) Time-lapseshapes along the movement. (Right) Color-coded space–time maps of cur-vature Cthðs; tÞ. (A) Graviceptive A model where the response only dependson the local angle. As the organ approaches the vertical, the basal partcontinues to curve. The organ overshoots the vertical, and the number ofoscillations increases with time (Movie S3). (B) Graviproprioceptive ACmodelwhere the response also depends on the local angle and the local curvature.Here the curvature decreases before reaching the vertical. It does exhibit anS shape, but oscillations are dampened, and the organ converges to a solu-tion where the curvature is focused near the base (Movies S4 and S5).

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the orientation of microtubules that may then modify therate of differential growth (28, 29).

This hypothesis yields a model called the “graviproprioceptive”model, or the “AC model”:

∂Cðs; tÞ=∂t = −βAðs; tÞ− γCðs; tÞ; [5]

in the growth zone (i.e., for s>L−Lgz), and 0 elsewhere. Herethe change in curvature is directly related to the local curvatureitself via the parameter γ, the proprioceptive sensitivity. A moresystematic derivation of the A and AC models from symmetryarguments and rod kinematics is given in SI Appendix. The solu-tion of the AC model has the form:

Aðs; tÞ = A0e−γtP∞n=0

�βsγ2t

�−n=2Jn�2

ffiffiffiffiffiffiβts

p �

=A0e−βs=γ −A0P∞n=1

ð−1Þn�βsγ2t

�n=2Jnð2

ffiffiffiffiffiffiβts

p Þ;[6]

where it is seen that the dependence onffiffiffiffits

pand

ffiffiffiffiffiffit=s

pis retained,

but there is now an infinite sequence of Bessel functions. Thefirst of the two expressions is appropriate for short times. Thelatter is appropriate for long times and shows that the oscil-lations are now dampened toward a final steady state, whoseform is:

Af ðsÞ = A0e−βs=γ = A0e−s=Lc : [7]

The dynamics of the AC model (Fig. 3B and Movies S4 andS5) is now qualitatively consistent with the experiments: theoscillations are dampened, and the organ converges to a steadystate where the curvature is focused near the base through atypical GC/BS biphasic pattern.The convergence length Lc = γ=β is given by the decay length of

the exponential toward the vertical, and it results from the bal-ance between graviception and proprioception. The AC modelthus gives a direct explanation of the common BS (autotropic)phase, where curvature starts to decrease before reaching thevertical (7, 20). For purely geometrical reasons (lever-arm effect,Eq. 1), the apical angles decrease faster than the basal angles.Thus, curvature sensing first takes over gravisensing at the tip anddecurving starts there. It then moves downward together with thedecrease of A without any need for a systemic basipetal propa-gative signal. Another important scale is Lgz, the effective lengthof the growth zone where active curving can be achieved. Theratio Bl =Lgz=Lc = βLgz=γ is a dimensionless number that controlsimportant aspects of the dynamics.To assess whether the organ has time to converge to a steady

state before the apex crosses the vertical, thereby avoidingovershooting, the time of convergence Tc can be compared withthe time required for the apex to first reach the vertical, Tv.Using Eq. 5, Tc can be approximated from the proprioceptiveterm that dominates when approaching convergence as Tc = 1=γand Tv can be approximated as Tv = 1=ðβLgzÞ from the grav-iceptive term dominating initial dynamics. This gives a “tempo-ral” dimensionless number Bt =Tc=Tv = βLgz=γ, which is actuallyidentical to Bl. The fact that Bt =Bl gives a direct link betweenconvergence timing, transient modes, and steady-state form (i.e.,a kind of form-movement equivalence). We call this number the“bending number” denoted by B.To compare theory and experiments, B, Lgz, and Lc were

measured morphometrically from initial and steady-state imagesas shown for Arabidopsis inflorescence in Fig. 4. Because Lgz isthe length of the organ that has curved during the experiment, itcan be directly estimated by comparing the two images. By def-inition, Lc can be measured directly on the image of the finalshape as the characteristic length of the curved part (Fig. 4). Thebending number B ranged from around 0.9–9.3 displaying broad

intraspecific and interspecific variability over the experiments.Therefore, the AC model can be assessed from them.The kinematic data from wheat, Arabidopsis, and poplar was

analyzed in more detail to track the tropic movement after tilting(Fig. 1). The analytical solution Aacðs; tÞ for the AC model (Eq.6) was compared with the experimental angle space-time maps,given the bending number value. Angles were chosen instead ofcurvature here, as otherwise the determination of curvaturewould involve a derivative, producing more noise. The initialvalue of B for parameter estimation was estimated morpho-metrically. As the AC model does not account for elongationgrowth, we trimmed the data for wheat and Arabidopsis to thelength of the growth zone at the beginning of the experiment, asshown in Fig. 5. Typical results from Arabidopsis infloresencesare shown in Fig. 5, and additional results from Arabidopsis,wheat, and poplar are provided in SI Appendix, Figs. S6, S7, andS8, respectively. The AC model was found to capture the com-mon features of the angle space-time maps over the entire GC/BS process (compare Fig. 5 A and B). The (dimensionless) meanslope of comparison of the model vs. data (for the three speciestogether) was 1.00 ± 0.15, the intercept was 0.07 ± 0.20, and thecoefficient of determination was 0.92 ± 0.05, so the AC modelcaptured around 90% of the total experimental variance inAðs; tÞ and displayed no mean quantitative bias.The form–movement equivalence predicted by the AC model

was then directly assessed through a simple morphometric anal-ysis of the tilting experiments on the 12 angiosperm genotypes.More precisely, we assessed whether the AC model predicted thediscrete transitions between transient oscillatory modes aroundthe vertical (e.g., Fig. 1 and SI Appendix, Fig. S5) with increasingvalues of the bending number B. At a given time t, the currentmode is defined as the number of places below the apex wherethe tangent to the central line of the organ is vertical (SI Ap-pendix, Fig. S5). If there is one vertical tangent more basal thanthe apex, then the organ overshoots the vertical once. This ismode 1, when a C shape is formed. If an S shape develops, thenthe transient mode will be mode 2, and a Σ shape is mode 3, andso on. The mode number M of the whole movement is then givenby the maximal mode of all of the transitory shapes (e.g., in SIAppendix, Fig. S5, the mode of the movement of the inflorescenceisM = 1 as a transient C shape is seen but not an S shape). In Fig.6, the modes of 12 plant organ responses were plotted against therespective estimated bending numbers and compared with thepredictions of the AC model.

0 10 20 30 40 50

0

0.5

1

1.5

s(mm)

A

A B

Fig. 4. Morphometric measurement of the bending number B from steady-state configurations of Arabidopsis inflorescences. (A) Estimation of the ef-fective length Lgz by superimposing the first and last kinematics images. Thered dotted lines indicate the zone where the organ started to curve. Theeffective length of the organ can then be defined as the distance from thispoint to the apex of the initial plant on the first image. (B) Estimation of theconvergence length Lc by plotting the local inclination angle Aðs; tÞ alongthe organ beginning from the curved zone. To extract the convergencelength Lc , the angle Aðs; tÞ is fitted with the exponential Aðs; tÞ=A0e−s=Lc , n =28, R2 = 0.99.

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The prediction displays stepwise increases in modes at bendingnumbers corresponding to 2.8 for the transition from mode 0 tomode 1 and 3.9 for the transition from mode 1 to mode 2. Noplant in the experiments displayed mode transitions for smallerbending numbers than was predicted by the AC model. Manyindividual plant responses were found near the transition frommode 0 to mode 1—that is, between the mode in which theycannot reach the vertical and the mode where they overshoot thevertical and oscillate. The transition from mode 2 to mode 3 onlyoccurs for very large bending numbers (B> 10) and was neverseen in any of the experiments. In two-thirds of the plants, theprediction of the oscillations by the AC model was correct.However, about one-third of the plants oscillated less than pre-dicted. To some extent, this may be due to inaccuracies in theestimation of bending numbers, but second-order mechanisms(possibly related to elongation growth) are likely to be involved,ones that add to the common graviproprioceptive core describedby the AC model.Nevertheless, the fact that the AC model accounts for the

common GC/BS pattern with no quantitative bias and capturesthe transitions between three different modes over one order ofmagnitude of bending numbers and a broad taxonomical range isan indication of its robustness. All this strongly suggests thathypothesis 5 and its mathematical description by the AC modelcaptures the universal core of the control over gravitropic dy-namics. The longstanding sine law for gravitropism (3) shouldthus be replaced by the graviproprioceptive dynamic AC model,which highlights the equal importance of curvature- and grav-isensing. Doing so has already yielded three major insights.

i) The AC model can achieve distinct steady-state tip angles forthe same vertical GSA. In particular, plants with B< 2:8 can-not reach their GSA (as specified in the gravitropic term of

the AC model) even in the absence of biomechanical andphysiological limits in their motor bending capacity (3, 10,12). Therefore, the GSA cannot be measured directly fromexperiments and can only be assessed by AC model–assistedphenotyping.

ii) The fact that most plants display very few oscillations beforeconverging to the steady state despite destabilization throughlever-arm effects does not actually require the propagation oflong-distance biological signals and complex regulation. Thevalue of the dimensionless bending number simply has to beselected in the proper range—that is, graviceptive and pro-prioceptive sensitivities have to be tuned together as a func-tion of organ size possibly pointing to molecular mechanismsyet to be discovered.

iii) The AC model can account for the behavior of actively elon-gating organs despite neglecting the effects of mean elon-gation growth. Subapical elongation growth may have desta-bilizing effects by spreading curvature, convecting, and fixingit outside the growth zone in mature tissues (14). Our resultmeans that the values for the time of convergence to thesteady-state Tc were small enough compared with the charac-teristic times for elongation growth in all of the species studied.As Tc depends mostly on the proprioceptive sensitivity,possibly there is natural selection for this trait as a functionof the relative elongation rate (and organ slenderness) andfor fine physiological tuning.

Proprioceptive sensing is thus as important as gravisensing forgravitropism. The study of molecular sensing mechanisms (2, 15)can thus now be extended to the cross-talk between gravi- andpropriosensing as a function of organ size. Candidate mechanismsfor the proprioception of the curvature may involve mechanicalstrain- or stress-sensing (27, 30) triggering microtubules reor-ientation (28, 29). Ethylene seems to be involved (17) but not thelateral transport of auxin (21). Whatever the detailed mechanismsinvolved, putative models of molecular networks controllinggraviproprioceptive sensing (31) should be consistent with the ACmodel and with the existence of a dimensionless control param-eter, the bending number. Moreover, the bending number B is areal quantitative genetic trait (32, 33). It controls the whole dy-namics of tropic movement and encapsulates both the geometry

s(mm)

t(h

)

0 20 40 60

0

5

10

15

20

0

0.5

1

1.5

s(mm)

t(h

)

0 20 40 60

0

5

10

15

20

0

0.5

1

1.5

−0.5 0 0.5 1 1.5−0.5

0

0.5

1

1.5

Ath

Aex

p

A B

C

Fig. 5. Quantitative comparison between experimental (exp) and predicted(th) angle space–time maps of Aðs; tÞ for an Arabidopsis inflorescence for thewhole gravitropic response. (A) Experimental angle space–time map ofAexpðS; tÞ trimmed for s≤ Lðt =0Þ, as the ACmodel does not consider changesin length. (B) Angle space–time map predicted by the AC model Aacðs; tÞ. (C)Quantitative validation plot of experimental Aexpðs; tÞ vs. theoreticalAthðs; tÞ. Orthogonal linear fit slope, 1.13; intercept, 0.17; R2 = 0.94.

0 1 2 3 4 5 6 7 8 9

0

1

2

B

mo

de

model ACwheat coleoptile (Recital)bean hypocotylesunflower hypocotylepea epicotyletomato plant stemchili plant stemraspberry bush stemImpatiens glandilufera stem (Pfeffer)carnation inflorescenceAt inflorescence (Col0)At inflorescence (Col0 pin1 mutant)Hybrid populus (deltoides x nigra) (Coutand et al. 2007))

Fig. 6. Mode number M plotted against bending number B= Lgz=Lc for in-dividual plants (N = 67). The green line shows the same plot for theACmodel.

Bastien et al. PNAS Early Edition | 5 of 6

PLANTBIOLO

GY

PHYS

ICS

and the perception–regulation functions involved (34). The sim-ple measurement of B is now possible and this may be used forthe high-throughput phenotyping of mutants or variants in manyspecies. From a more general perspective, it would now be in-teresting to explore how plants manage to control gravitropismdespite the destabilizing effects of elongation growth. Areas toinvestigate are whether there is physiological tuning of B duringgrowth and whether there is natural selection for proprioceptivesensitivity as a function of the relative elongation rate and organslenderness. For this, it will be necessary to combine noninvasivekinematics methods to monitor elongation growth at the sametime as curvature (e.g., refs. 32, 33) with a more general modelthat explicitly includes the expansion and convection of cellsduring growth (3, 14). Finally, this approach can also be used tostudy the gravitropism of other plant organs and other growthmovements like phototropism or nutation, which will showwhether this theory of active movement is universal.

Materials and MethodsExperiments were conducted in growth cabinets for etiolated wheat coleoptiles(Triticum aestivum cv. Recital) or controlled temperature greenhouses for thenine other types of plant organs—bean hypocotyl (Phaseolus vulgaris), sun-flower hypocotyl (Helianthus annuus), pea epicotyl (Pisum sativum), tomatostem (Solanum lycopersicum), chili stem (Capsicum annuum), raspberry cane(Rubus ideaus), carnation inflorescence (Dianthus caryophyllus), and Arabidopsisthaliana inflorescences from a wild-type (ecotype Col0) and its pin1 mutant[a mutant of the PIN1 auxin efflux carier displaying reduced auxin longitudinal

transport (11) (see SI Appendix, sections S2.1 and S2.4 for more details)]. Plantswere grown until a given developmental stage of the organ of interest (e.g.,until the beginning of inflorescence flowering for Arabidopsis in Fig. 4). Theywere then tilted and clamped horizontally A(s = 0, t) = ϕ/2 for all t underconstant environmental conditions in the dark (to avoid interactions withphototropism). Number of replicates were 30 for wheat, 15 for Arabidopsis, and5 for all the other species. Published data were also reprocessed from similarexperiments on Impatiens glandilufera stems by Pfeffer (35) and on poplartrunks (Populus deltoides x nigra cv I4551) by Coutand et al. (9). More precisely,two types of experiments were conducted, as explained in SI Appendix, sectionS2.2 and S2.5: (i) detailed kinematics experiments on two model species(Arabidopsis and wheat), based on time-lapse photography and quanti-tative analysis of curving-decurving kinematics (SI Appendix, sections S2.2to S2.4) and (ii) simplified morphometric experiments on all the genotypes, toestimate the bending number (through Bl = Lgz/Lc) and the (transient) globalmode M, defined as the maximum number of places below the apex wherethe tangent to the central line of the organ is vertical (SI Appendix, Fig. S5and section S2.5). Quantitative assessment of the AC model was conductedby fitting Eq. 6 to the datasets from the detailed kinematics experiments(including also poplar; see SI Appendix, section S2.6), whereas a qualitativeassessment on mode transitions and space-time equivalence was conductedon the dataset from the morphometric experiment (including also Impatiens;see SI Appendix, section S2.5).

ACKNOWLEDGMENTS. We thank Dr. C. Coutand for providing the poplardata, S. Ploquin and Dr. C. Girousse for help with the wheat experiments,Drs. A. Peaucelle and H. Hofte for help with the Arabidopsis experiments,and Emondo (Boston) for editing the English.

1. Darwin C (1880) The Power of Movements in Plants (D. Appleton and Company, New

York).2. Gilroy S, Masson PH (2008) Plant Tropisms (Blackwell, Oxford).3. Moulia B, Fournier M (2009) The power and control of gravitropic movements in

plants: A biomechanical and systems biology view. J Exp Bot 60(2):461–486.4. Moulia B, Coutand C, Lenne C (2006) Posture control and skeletal mechanical accli-

mation in terrestrial plants: Implications for mechanical modeling of plant architec-

ture. Am J Bot 93(10):1477–1489.5. Skotheim JM, Mahadevan L (2005) Physical limits and design principles for plant and

fungal movements. Science 308(5726):1308–1310.6. Galland P (2002) Tropisms of Avena coleoptiles: Sine law for gravitropism, exponen-

tial law for photogravitropic equilibrium. Planta 215(5):779–784.7. Firn RD, Digby J (1979) A study of the autotropic straightening reaction of a shoot

previously curved during geotropism. Plant Cell Environ 2(2):149–154.8. Stockus A, Moore D (1996) Comparison of plant and fungal gravitropic responses

using imitational modelling. Plant Cell Environ 19(7):787–800.9. Coutand C, Fournier M, Moulia B (2007) The gravitropic response of poplar trunks:

Key roles of prestressed wood regulation and the relative kinetics of cambial growth

versus wood maturation. Plant Physiol 144(2):1166–1180.10. Forterre Y, Skotheim JM, Dumais J, Mahadevan L (2005) How the Venus flytrap snaps.

Nature 433(7024):421–425.11. Noh B, Bandyopadhyay A, Peer WA, Spalding EP, Murphy AS (2003) Enhanced gravi-

and phototropism in plant mdr mutants mislocalizing the auxin efflux protein PIN1.

Nature 423(6943):999–1002.12. Bennett MJ, Roberts I, Palme I (2000) Moving on up: Auxin-induced K+ channel ex-

pression regulates gravitropism. Trends Plant Sci 5(3):85–86.13. Goriely A, et al. (2008) Elastic growth models.Mathematical Modelling of Biosystems,

ed Mondaini R (Springer-Verlag, Berlin and Heidelberg), pp 1–44.14. Silk WK (1984) Quantitative descriptions of development. Annu Rev Plant Physiol 35:

479–518.15. Blancaflor EB, Masson PH (2003) Plant gravitropism. Unraveling the ups and downs of

a complex process. Plant Physiol 133(4):1677–1690.16. Morita MT (2010) Directional gravity sensing in gravitropism. Annu Rev Plant Biol

61(1):705–720.17. Pickard BG (1985) Roles of hormones, protons and calcium in geotropism. Encyclo-

pedia of Plant Physiology, eds Phais RP, Reid DM (Springer, Berlin), Vol III, pp 193–281.18. Stankovic B, Volkmann D, Sack FD (1998) Autotropism, automorphogenesis, and

gravity. Physiol Plant 102(2):328–335.19. Meskauskas A, Moore D, Novak Frazer L (1998) Mathematical modelling of mor-

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room stem of Coprinus cinereus. New Phytol 140(1):111–123.

20. Meskauskas A, Novak Frazer L, Moore D (1999) Mathematical modelling of mor-phogenesis in fungi: A key role for curvature compensation (“autotropism”) in thelocal curvature distribution model. New Phytol 143(2):387–399.

21. Haga K, Iino M (2006) Asymmetric distribution of auxin correlates with gravitropismand phototropism but not with autostraightening (autotropism) in pea epicotyls. JExp Bot 57(4):837–847.

22. Kuznetsov OA, Hasenstein KH (1997) Magnetophoretic induction of curvature incoleoptiles and hypocotyls. J Exp Bot 48(316):1951–1957.

23. Firn RD, Digby J, Hall A (1981) The role of the shoot apex in geotropism. Plant CellEnviron 4(2):125–129.

24. Digby J, Firn RD (1995) The gravitropic set-point angle (GSA): The identification of animportant developmentally controlled variable governing plant architecture. PlantCell Environ 18(12):1434–1440.

25. Sherrington CS (1907) On the proprio-ceptive system, especially in its reflex aspect.Brain 29(4):467–482.

26. Coutand C, Moulia B (2000) Biomechanical study of the effect of a controlled bendingon tomato stem elongation: Local strain sensing and spatial integration of the signal.J Exp Bot 51(352):1825–1842.

27. Moulia B, et al. (2011) Integrative mechanobiology of growth and architectural de-velopment in changing mechanical environments. Mechanical Integration of PlantCells and Plants, ed Wojtaszek Springer P (Springer, Berlin), pp 269–303.

28. Fischer K, Schopfer P (1998) Physical strain-mediated microtubule reorientation in theepidermis of gravitropically or phototropically stimulated maize coleoptiles. Plant J15(1):119–123.

29. Ikushima T, Shimmen T (2005) Mechano-sensitive orientation of cortical microtubulesduring gravitropism in azuki bean epicotyls. J Plant Res 118(1):19–26.

30. Hamant O, et al. (2008) Developmental patterning by mechanical signals in Arabi-dopsis. Science 322(5908):1650–1655.

31. Rodrigo G, Jaramillo A, Blázquez MA (2011) Integral control of plant gravitropismthrough the interplay of hormone signaling and gene regulation. Biophys J 101(4):757–763.

32. Miller ND, Parks BM, Spalding EP (2007) Computer-vision analysis of seedling re-sponses to light and gravity. Plant J 52(2):374–381.

33. Brooks TL, Miller ND, Spalding EP (2010) Plasticity of Arabidopsis root gravitropismthroughout a multidimensional condition space quantified by automated imageanalysis. Plant Physiol 152(1):206–216.

34. Coen E, Rolland-Lagan AG, Matthews M, Bangham JA, Prusinkiewicz P (2004) Thegenetics of geometry. Proc Natl Acad Sci USA 101(14):4728–4735.

35. Pfeffer WTG (1898–1900) Kinematographische Studien an Impatiens, Vicia, Tulipa,Mimosa und Desmodium [Kinematics Studies of an Impatiens, Vicia, Tulipa, Mimosaand Desmodium] (Timelapse Photography, Color: No, Sound: No, 3min 30) (UniversitätLeipzig, Botanisches Institut, Leipzig, Germany). Video transcription by Kinescope.Available at www.dailymotion.com/video/x1hp9q/. Accessed November 20, 2012.

6 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1214301109 Bastien et al.

A unifying model of shoot gravitropism revealsproprioception as a central feature of posturecontrol in plants - supplementary information-Renaud Bastien ∗ † ‡, Tomas Bohr §, Bruno Moulia ∗ † ¶ and St ephane Douady ‡ ¶

∗INRA, UMR547 PIAF Physique et physiologie Intégratives de l’Arbre fruitier et Forestier, F-63100 Clermont-Ferrand, France ,†Clermont Université, Université BlaisePascal, UMR547 PIAF Physique et physiologie Intégratives de l’Arbre fruitier et Forestier , BP 10448, F-63000 Clermont-Ferrand, France,‡Matiere et SystemesComplexes, Universite Paris-Diderot, 10 rue Alice Domont et Leonie Duquet, 75025 Paris Cedex 13, France,§Department of Physics and Center for Fluid Dynamics,Technical University of Denmark, DK-2800 Lyngby, Denmark, and ¶These authors have contributed equally and complementary to the work

1- Construction of the mathematical model1.1 General equation. The plant organs considered in this study areslender structures. During tropic movements cells do not undergoshear growth and torsion can be neglected (3). Therefore the organcan be considered mechanically as an (actively) flexing rod (3). Itssuccessive shapes can thus be fully described by the local orientationA(s, t) and the local curvatureC(s, t) fields. Note that curvatureC is an objective quantity defining the local shape of the organ irre-spective of its local inclinationA. The mechanism that produces themovement, like differential growth, modifies the local curvature ofthe organ. The equation that drives the system should thus determinethe temporal variation in the local curvature

∂C(s, t)

∂t= φ, [1]

whereφ is a function of the geometry, the biomechanics of bendingand the perception-regulation process,s is the curvilinear abscissafrom the base to the apex andt is the time elapsed since the plant wastilted horizontally. It is postulated that the perception-regulation pro-cess driving the dynamics of the movement is of the first order, i.e.that the biomechanical motors are not limiting the movement as is of-ten the case (3). In addition, the perception involved in gravitropismis local (see the argument for Hypothesis H1 in the main text). Theperception of a segment at positions should be a function of the lo-cal angle and curvatureA(s, t) andC(s, t), and this local perceptionthen results in a local response. Equation [1] can thus be rewritten

∂C(s, t)

∂t= φ(A,C). [2]

Assuming that both the tilting angleA(s, t) and the curvatureC(s, t) are small, the functionφ can be expanded as polynomials ofA(s, t) andC(s, t) near the vertical (straight) configurations of theorgan.

∂C(s, t)

∂t= α+ β1A+ β2A

2 + ...+ γ1C + γ2C2 + ...+

+δ1AC + δ2A2C + δ3AC2 + ... [3]

When the organ is nearly straight and vertical, there is no gravitropicresponse. SoA = 0, C = 0 is a stable solution of the equation.Furthermore, as the behavior of the organ is independent of rotationaround the vertical axis, the transformationA → −A, C → −Cshould leave the system unchanged. This implies that all even-orderterms in [3] disappear, yielding

∂C(s, t)

∂t= β1A+ β3A

3 + ...+ γ1C + γ3C3 + ...

+δ2A2C + δ3AC2 + ... [4]

This is the most general equation describing gravitropism of an elon-gated aerial organ.

For simplicity, we will first assume that theβ andγ coefficientsdo not depend on positions or time t, i.e. that the sensitivities toangle and curvature are both time-independent and homogeneous.These assumptions have experimental support. Time-independenceof the straightening response is envisageable in that tropic responsesare fast in terms of the entire developmental timecourse of the organ(S1). Spatial homogeneity of the sensing capacity throughout thegrowth zone is supported by observations of the even distribution ofstatocytes or the response to high magnetic fields (22).

1.2 Test of two response functions: sine law (the A model ) and ex-ponential law. We may now compare two phenomenologicalφ func-tions that have been proposed in the literature, the sine law and theexponential law, to the general equation [4].

The sine law was defined by Sachs in the 19th century (see (3)for a review). Here then equation [2] can be rewritten as

∂C(s, t)

∂t= α sinA(s, t) [5]

whereα is a parameter. Expandingsin(A) as a power series (validfor anyA) yields

sinA = A−A3

3!+

A5

5!+O(A7) [6]

Since there are no even-order terms, the equation satisfies the sym-metry condition mentioned above and the sine law is thus a specialinstance of equation [4]. In this work, we have used the approxima-tion sinA ≈ A, for equation [5] giving

∂C(s, t)/∂t = −βA(s, t) [7]

Equation [7] is a mathematical specification of the hypothesisthat the rate of local change in local curvatureC is controlled only bya graviceptive term depending on the local inclination angleA. Wehave thus called this model the graviceptive model, or theA model(see also equation [3] in the main text).

We may now consider the exponential law postulated in the com-plete model of the tropic reaction in (20). This law is described bythe following function

∂C(s, t)

∂t= α e

A−π/2A1 = α e

AA1 e

− π2A1 [8]

The effect of this function becomes very small whenA is lessthanπ/2 − A1 and thus only affects the start of the reaction. Thepower series ofeA/A1 is given by

eA/A1 = 1 +A

A1

+1

2

(

A

A1

)

2

+1

3!

(

A

A1

)

3

+1

4!

(

A

A1

)

4

+ ... [9]

Here even terms appear so this function violates the symmetries ofthe system and is therefore not a suitable model. This illustrates theimportance of considering the symmetry of the problem when mod-eling especially when the exponential function is used.

PNAS 1–11

1.3 First-order equation. NearA(s, t) = 0 andC(s, t) = 0, equa-tion [4] can be linearized. This first order expression is in fact asecond order approximation since we have seen previously that allthe even terms disappear and the first cross-product termsA2C andC2A are third-order. The generalized equation of gravitropism at thesecond order is thus given by

∂C(s, t)

∂t= −βA− γC [10]

For an initially straight organ clamped at the base and tilted with aninitial angleA0 from the vertical, the boundary conditions are thengiven by

A(0, t) = A(s, 0) = A0 [11]

C(s, 0) = 0 [12]

Equation [10] is a mathematical specification of the hypothesis thatthe rate of change in local curvatureC is controlled by a gravicep-tive term depending on local inclination angleA and a proprioceptiveterm depending on the sensing of local curvature by each organ seg-mentC (while respecting the symmetry of the problem, and usinga second order approximation). This new model has been thereforecalled the graviproprioceptive model, or theAC model (see also equa-tion [5] in the main text).

1.4 Steady state and the dimensionless number Bl. The steadystate of the equation [10] is given by

∂C(s, t)

∂t= 0 [13]

−βA(s, t)− γC(s, t) = 0 [14]

C(s, t) = ∂sA(s, t) [15]

As A(s, t) = 0 is forbidden by the boundary condition ifA0 6= 0, there is only one steady solution

−βA(s, t)− γ∂A(s, t)

∂t= 0 [16]

A(s, t) = A0e−

βsγ [17]

0 2 4 6 8 10

0

2

4

6

8

10

x

y

Fig. 1. Final shape of the AC model for different values of Bl with Lgz = 10.From the lower line (green) to the upper one (yellow) the values of Bl are re-spectively 0.5, 1, 2, 4 and 8.

Equation [17] thus defines the steady-state shape of the organ. Alongthe organ at steady state, the angleA(s) decreases fromA0 to A0/e(∼ 0.37A0) over a lengthLc given by

LC =γ

β[18]

Lc is called the convergence length. It is then possible to designate adimensionless numberBl by expressingLc relatively to a character-istic effective length for bendingLgz (the length of the growth zonewhere active curving can be achieved):

Bl =βLgz

γ[19]

Each value ofBl corresponds to one and only one specific shape(Figure 1). WhenBl is a small number the apex of the organ cannotreach the vertical despite the fact that the graviceptive setpoint angleis A = 0, because the convergence length is too large compared tothe length of the organ.

1.5 Timing of the movement and dimensionless number Bt. It isinsightful to compare the time for the apex to reach the vertical tothe time for the organ to converge to its final shape. Indeed when theorgan reaches the vertical some time before convergence occurs, theorgan may exhibit transient spatial oscillations.

A straightforward (under)estimation of the time for the apex toreach the vertical can be obtained by ignoring the proprioceptive pro-cess and further assuming that the angleA stays at its maximal valueof A0.

∂C(s, t)

∂t≈ −βA0 [20]

with the solution

A(s, t) = A0 −

∫ s

Lgz

dsβA0t [21]

i.e.,

A(0, t) = A0 −

∫

0

Lgz

dsβA0t = A0(1− βLgzt) [22]

Thus the timeTv to bring the apex to the vertical orientation (A = 0)is

Tv =1

βLgz[23]

Likewise, when the graviproprioceptive term dominates, the conver-gence timeTc to the final shape is given by the characteristic timerequired by the organ to reach the steady state

Tc =1

γ[24]

It is now possible to define a dimensionless number for the movementas the ratio of the convergence timeTc and the vertical timeTv

Bt =βLgz

γ[25]

By comparing equation [19] and equation [25] we see

Bl = Bt = B [26]

This "bending number" will quantify the number of transient over-shoots that occur when the organ approaches the steady state as dis-cussed in the main text.

2

s

t

0 5 10

0

1

2

3

4

5

0

0.5

1

1.5

st

0 5 10

0

1

2

3

4

5

0

0.5

1

1.5

−1 0 1 2−1

−0.5

0

0.5

1

1.5

2

Aa

As

Fig. 2. Quantitative comparison between the analytical solution Aa(s, t) (left panel) for the angle and the numerical solution As(s, t) (middle panel) for the A modelwith β = 1 and Lgz = 10. Quantitative validation plot Aa(s, t) vs As(s, t) (right panel) with orthogonal linear fit (slope 1.0, intercept 0.0, R2 = 1.0).

s

t

0 5 10

0

1

2

3

4

5

0

0.5

1

1.5

s

t

0 5 10

0

1

2

3

4

5

0

0.5

1

1.5

−0.5 0 0.5 1 1.5 2−0.5

0

0.5

1

1.5

2

Aa

As

Fig. 3. Quantitative comparison between the analytical solution Aa(s, t) (left panel) of the angle and the numerical solution As(s, t) (middle panel) for the AC modelwith B = 10 and Lgz = 10. The quantitative validation plot Aa(s, t) vs As(s, t) (right panel) with orthogonal linear fit (slope 1.0, intercept 0.0, R2 = 1.0)

1.6 Analytical Solution and Numerical Simulations. The A modelcorresponds to the case, where the proprioceptive term is removed,and only the angle perception is kept:

∂C(s, t)

∂t= −βA [27]

With the initial conditionA(s, t = 0) = A0 this has the solution

A(s, t) = A0

√

βt

sJ0

(

2√

βts)

[28]

as can be seen by directly inserting it into the equation and perform-ing the differentiations (S2) using

C(s, t) =∂A(s, t)

∂s[29]

This analytical solution A(s, t) of theA model [28] was comparedto the angle space maps obtained through numerical simulations ofEquation [27], for many sets of values for the two parameters. Atypical example is shown in Figure S3. Again, no discrepancies werefound between the analytical solution and the numerical experiments,so Equation 28 is correct and can be used to investigate the behaviorof theA model and assess it against experimental data.

The analytical solution of the graviproprioceptive equation [10]with boundary conditions [11] and [12] are

A(s, t) = A0e−γt

∞∑

n=0

γn/2

(

βs

γt

)−n/2

Jn

(

2√

βts)

[30]

which can also be verified by direct differentiation, although morecumbersome (S2). This analytical solution A(s, t) of theAC model[30] was compared to the angle space map obtained through numeri-cal simulations of Equation [10] for many sets of values for the two

parameters. A typical example is shown in Figure S2. No discrep-ancies were found between the analytical solution and the numericalexperiments, so Equation [30] is correct and can be used to investi-gate the behavior of theAC model and assess it against experimentaldata. The detailed mathematical derivation of the analytical solutionsis available on ArXiv (S2).

PNAS 3

2- Experiments2.1 - Plant Materials and tilting experiments. Experiments were con-ducted in growth cabinets for etiolated wheat coleoptiles (Triticumaestivum cv. Recital) or controlled temperature greenhouses forthe nine other types of plant organs - bean hypocotyl (Phaseolusvulgaris), sunflower hypocotyl (Helianthus annuus ), pea epicotyl(Pisum sativum), tomato stem (Solanum lycopersicum), chili stem(Capsicum annuum ), raspberry cane (Rubus ideaus), carnation in-florescence (Dianthus caryophyllus ), andArabidopsis thaliana in-florescences from a wild type (ecotype Col0) and itspin1 mutant(a mutant of the PIN1 auxin efflux carier displaying reduced auxinlongitudinal transport (11)). Plants were grown until a given devel-opmental stage of the organ of interest (e.g until the beginning ofinflorescence flowering for Arabidopsis in Figure 4). They were thentilted and clamped horizontallyA(s = 0, t) = π/2 for all t un-der constant environmental conditions in the dark (to avoid interac-tions with phototropism). Number of replicates were 30 for wheat, 15for Arabidopsis and 5 for all the other species. Published data werealso reprocessed from similar experiments onImpatiens glandiluferastems by Pfeffer (35) and on poplar trunks (Populus deltoides x nigracv I4551) by Coutand et al. (9).

2.2 - Detailed kinematics experiments. Time lapse photography wasperformed using a flash light, where the light was filtered to retainonly green light, which did not stimulate any phototropic response.After initial tilting of the organ, the tropic movement was followeduntil a clear steady-state shape was achieved. One typical experimenton Arabidopsis thaliana is presented in Figure S5.A.

2.3 - Kinematics of Curving-Decurving. The central line of the organwas extracted from the pictures at successive timest and curvatureCand curvilinear abscissas at successive points along the central linewere computed using a method described in refs (3,9). Space-timeplots with color coding indicating the magnitude of the angle or thecurvature were then generated to illustrate the pattern of the gravit-ropic movement.

2.4 Plant Material: phylogenetics of species studied. Elevenspecies were chosen from a broad taxonomical range of land an-giosperms including monocots and dicots, see Figure S4. Differenttypes of organs were studied: coleoptile, hypoctyl, epicotyl, inflo-rescence stems, or stems of vegetative shoots. The plants studiedrepresent many of the growth habits of angiosperms, e.g. herbaceousplants, biennial shrubs and trees.

4

Fig. 4. Phylogenetic distribution of the species under study (modified from The Angiosperm Phylogeny (S3)). Families of the studied species are marked by a red dot.

PNAS 5

2.5 Morphometric experiment and Characterization of transi entoscillatory modes. Estimates of the bending number (throughBl =Lgz/Lc) and of the (transient) global modeM (Figure S5.B) wereobtained. To estimate the lengthLgz (the length of the organ alongwhich active curving can take place), the first image of the kinemat-ics just after tilting the organ was compared to the last one when theorgan has reached a steady state (Figure 4). The distance betweenthe apex and the most basal point with non-zero curvature on the lastimage gives the total length that was able to curve at the start of theexperiment.This gives an approximation of the length of the growthzoneLgz .To get the convergence lengthLc on the image of the steadystate shape, the local orientation angle is taken from the point wherethe organ started to curve (Figure 4). Then the plot ofA(s) is fitted

with an exponential function,A0e−s/Lc . This fit gives a direct esti-

mate of the convergence length to the verticalLc. The measurementof the modes are illustrated in Figure S5.B. At a given timet the cur-rent mode is defined as the number of places below the apex wherethe tangent to the central line of the organ is vertical (Figure S5.B).In the example (Figure S5.A) the inflorescence of Arabidopsis dis-played transient J then C shape and finally, just before convergence,an S shape. (Figure S5.B). Transient oscillatory modes were char-acterized by the mode number M defined as the maximal number ofplaces in which the tangent to the central line of the organ is verticalsimultaneously during the straightening mouvement. This transientstate is the most curved state. In our experiments, only modes 0, 1and 2 were observed. The value of mode M for (Figure S5.A) tiltingexperiment is thus M= 2.

mode 0 mode 1 mode 2

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B.

Fig. 5. A.Timelapse photographs of a tilting experiment on the inflorescence of Arabidopsis thaliana taken at 2- hour intervals. The apical part overshoots the verticalonce 8 hours after tilting (4th image, C-shape). The white bar is 1 cm long. B.Quantification of the transient modes of the gravitropic movement. Mode M is themaximal number of places in which the tangent to central line of the organ crosses the vertical simultaneously. The curved line represents the gravitropic organ andthe dashed lines represent different modes. No dashed line, mode 0 or J-shape; one dashed line, mode 1 or C shape; 2 dashed lines, mode 2 or S shape.

6

2.6 Quantitative Assessment and Statistical Fit.. The analyticalexpression for the angleAac of the AC model [10] was fitted to theexperimental angle-space mapping numerically through a non lin-ear optimization algorithm combining steepest gradient with randomsampling of the parameter space (to avoid local minima), using thebending number estimated from the morphometric method as a start-ing value. The comparison between the measured angle dynamicsAexp(s, t) and that predicted by theAC modelAac(s, t) was basedon orthogonal functional linear regression, since the model predictioncan also display random errors through the estimation ofB (S4).

PNAS 7

3. Detailed kinematics experiments and quantitative assess mentof the AC model.

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Fig. 7. Quantitative comparison between experimental (exp)and predicted (th) angle spacetime map of A(s, t) in wheat coleoptile for the entire gravitropic reponse ofthree different individuals A, B and C. Experimental angle spacetime map of Aexp(s, t) (right panels), angle spacetime map predicted by the AC model Ath(s, t)(middle panels) and quantitative validation plot Aexp(s, t) vs Ath(s, t) (right panels). A. Orthogonal linear fit slope 1.16, intercept 0.0003, R2 = 0.97. B. Orthogonallinear fit slope 0.97, intercept 0.078, R2 = 0.96. C. Orthogonal linear fit slope 0.88, intercept 0.15, R2 = 0.96.

PNAS 9

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Fig. 8. Quantitative comparison between experimental (exp)and predicted (th) angle spacetime map of A(s, t) in poplar trunk for the whole straightening dynamicsof three individuals A, B and C. Experimental angle spacetime map of Aexp(s, t) (left panels), angle spacetime map predicted by the ACmodel Ath(s, t) (middlepanels) and quantitative validation plot Aexp(s, t) vs Ath(s, t) (right panels). A. Orthogonal linear fit slope 0.90, Intercept 0.037, R2=0.91, B. Orthogonal linear fitslope 0.88, Intercept 0.10, R2=0.86 and C. Orthogonal linear fit slope 0.80, Intercept 0.058, R2=0.80.

1. Silk WK, Erickson RO (1978) Kinematics of hypocotyl curvature. Am J Bot 65(3):310-319.

2. Bastien R, Moulia B, Douady S, Bohr T (2012) Analytical Solution of the Proprio-Graviceptive equation for shoot gravitropism of plants arXiv:1210.3480 [q-bio.TO]

3. The Angiosperm Phylogeny Group (2003) An update of the angiosperm phylogenygroup classification for the orders and families of flowering plants: Apg ii. BotanicalJournal of the Linnean Society 141(4):399-436.

4. Dagnelie, P (2006) Statistique theorique et appliquee. Tome 2. Infrence statistique aune et a deux dimensions (2nd ed). De Boeck et Larcier, Brussels, Belgium. French

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Supporting InformationBastien et al. 10.1073/pnas.1214301109

Movie S1. Gravitropic movement of a wheat coleoptile, after an initial tilting at 90° from the vertical. Note that this wheat coleoptile never overshot thevertical during the straightening process. Other coleoptiles in the experiment did not even reach the vertical even at their tip (not shown in the movie).

Movie S1

Bastien et al. www.pnas.org/cgi/content/short/1214301109 1 of 3

Movie S2. Gravitropic movement of an inflorescence of A. thaliana after an initial tilting at 90° from the vertical. The inflorescence of A. thaliana exhibiteda transient C shape during the straightening process and overshot the vertical.

Movie S2

Movie S3. Solution of the A model. The color (from blue to red) codes for the absolute value of the curvature C(s,t). The simulated organ never reachesa steady state and oscillation increases along the organ.

Movie S3

Bastien et al. www.pnas.org/cgi/content/short/1214301109 2 of 3

Movie S4. Solution of the AC model, B = 1. The color (from blue to red) codes for the absolute value of the curvature C(s,t). The simulated organ reachesa steady state but does not reach the vertical.

Movie S4

Movie S5. Solution of the AC Model, B = 10. The color (from blue to red) codes for the absolute value of the curvature C(s,t). The simulated organ reachesa steady state after exhibiting a transient S-shaped mode during the process.

Movie S5

Other Supporting Information Files

SI Appendix (PDF)

Bastien et al. www.pnas.org/cgi/content/short/1214301109 3 of 3